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How Does the Constant Product Formula Work in a Basic AMM?

The constant product formula, $x y = k$, ensures that the product of the quantities of the two tokens in the pool ($x$ and $y$) always remains equal to a constant ($k$). When a token is bought, its quantity decreases, and the other's increases, automatically adjusting the price to maintain the constant $k$.

How Does an Automated Market Maker (AMM) Algorithm Maintain the Constant Product in a Liquidity Pool?
How Does an Automated Market Maker (AMM) Work?
How Does the Constant Product Formula (X Y=k) Govern AMMs?
How Is the Price of a Token Determined within a Liquidity Pool?